How to understand mapping function of kernel?
A kernel function $f : \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}$ is a valid support vector kernel if it is a Mercer kernel. Mercer's condition essentially ensures that the Gram matrix of the kernel is positive semi-definite. Interestingly, this ensures that the SVM objective is convex.
The Euclidean distance function does not satisfy Mercer's condition since it's Gram matrix is not necessary positive semi-definite. Thus, it is not a valid kernel.